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On the entropy for semi-Markov processes

Part of: Communication, information Sufficiency and information Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Valerie Girardin  and
Nikolaos Limnios
Valerie Girardin*
Affiliation:
Université de Caen
Nikolaos Limnios*
Affiliation:
Université de Technologie de Compiègne
*
∗Postal address: Mathématiques, Campus II, Université de Caen, BP 5186, 14032 Caen, France. Email address: [email protected]
∗∗Postal address: Laboratoire de Mathématiques Appliquées, Université de Technologie de Compiègne, BP 20529, 60205 Compiègne Cedex, France.
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Abstract

The aim of this paper is to define the entropy of a finite semi-Markov process. We define the entropy of the finite distributions of the process, and obtain explicitly its entropy rate by extending the Shannon–McMillan–Breiman theorem to this class of nonstationary continuous-time processes. The particular cases of pure jump Markov processes and renewal processes are considered. The relative entropy rate between two semi-Markov processes is also defined.

Keywords

Relative entropyentropy ratesemi-Markov processpure jump Markov process

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes 60J25: Continuous-time Markov processes on general state spaces 62B10: Information-theoretic topics 94A17: Measures of information, entropy
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 1060 - 1068
Copyright
Copyright © Applied Probability Trust 2003 

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